Generalized limit theorem and bifurcation for problems with Pucci's operator
Keywords
Pucci's operator, bifurcation, one-sign solution, sign-changing solutionAbstract
We establish a new limiting result which extends the famous Whyburn's limit theorem. As applications, we study the existence and multiplicity of one-sign or sign-changing solutions with a prescribed number of simple zeros for the following problem \begin{equation*} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+\left(D^2 u\right)=\mu f(u) &\text{in } \Omega,\\ u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\mathcal{M}_{\lambda,\Lambda}^+$ denotes the Pucci extremal operator. Combining bifurcation approach with our generalized limit theorem, we determine the range of parameter $\mu$ in which the above problem has one or multiple one-sign or sign-changing solutions according to the behaviors of $f$ at $0$ and $\infty$, and whether $f$ satisfies the signum condition $f(s)s> 0$ for $s\neq0$.References
S. Alarcón, L. Iturriaga and A. Quaas, Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros, Calc. Var. 45 (2012), 443–454.
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rew. 8 (1976), 620–709.
A. Ambrosetti, R.M. Calahorrano and F.R. Dobarro, Global branching for discontinuous problems, Comment. Math. Univ. Carolin. 31 (1990), 213–222.
A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), 411–422.
C.J. Amick and R.E.L. Turner, A global branch of steady vortex rings, J. Reine Angew. Math. 384 (1988), 1–23.
D. Arcoya, J.I. Diaz and L. Tello, S-shaped bifurcation branch in a quasilinear multivalued model arising in climatoloty, J. Differential Equations 150 (1998), 215–225.
S.N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations 246 (2009), 2958–2987.
A. Bensoussan and J.L. Lions, Applications of Variational Inequalities in Stochastic Control, English transl., North-Holland Publishing Co., Amsterdam, New York, 1982.
H. Berestycki, On some nonlinear Sturm–Liouville problems, J. Differential Equations 26 (1977), 375–390.
J. Busca, M.J. Esteban and A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci’s operators, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 187–206.
X. Cabré and L. Caffarelli, Fully Nonlinear Elliptic Equation, American Mathematical Society, Colloquium Publication, vol. 43, 1995.
L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations II, Comm. Pure Appl. Math. 38 (1985), 209–252.
A. Cutri and F. Leoni, On the Liouville property for fully nonlinear equations, Ann Inst. H. Poicaré Anal. Non Linéaire 17 (2000), 219–245.
G. Dai, Two Whyburn type topological theorems and its applications to Monge–Ampère equations, Calc. Var. (2016), 55:97.
G. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. (2016), 55:72.
G. Dai, Eigenvalue, global bifurcation and positive solutions for a class of nonlocal elliptic equations, Topol. Methods Nonlinear Anal. 48 (2016), 213–233.
G. Dai, Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst. 36 (2016), 5323–5345.
D.G. de Figueiredo, J.P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003), 452–467.
D.G. de Figueiredo, P.L. Lions and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equation, J. Math. Pures et Appl. 61 (1982), 41–63.
P. Felmer and A. Quaas, Critical exponents for the Pucci’s extremal operators, C.R. Acad. Sci. Paris, Sér. I 335 (2002), 909–914.
P. Felmer and A. Quaas, On critical exponents for the Pucci’s extremal operators, Ann Inst. H. Poicaré Anal. Non Linéaire 20 (2003), 843–865.
P. Felmer and A. Quaas, Positive solutions to “semilinear” equation involving the Pucci’s operator, J. Differential Equations 199 (2004), 376–393.
P. Felmer, A. Quaas and B. Sirakov, Resonance phenomena for second-order stochastic control equations, SIAM J. Math. Anal. 42 (2010), 997–1024.
P. Felmer, A. Quaas and M. Tang, On uniqueness for nonlinear elliptic equation involving the Pucci’s extremal operator, J. Differential Equations 226 (2006), 80–98.
P. Felmer and A. Quaas, Some Recent Results on Equations Involving the Pucci’s Extremal Operators, Progress in Nonlinear Differential Equations and Their Applications, vol. 66, pp. 263–281.
P. Felmer, A. Quaas and B. Sirakov, Landesman–Lazer type results for second order Hamilton–Jacobi–Bellman equations, J. Funct. Anal. 258 (2010), 4154–4182.
G. Galise, F. Leoni and F. Pacella, Existence results for fully nonlinear equations in radial domains, Comm. Partial Differential Equations 42 (2017), 757–779.
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 8 (1981), 883–901.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 2001.
P. Korman, Y. Li and T. Ouyang, An exact multiplicity result for a class of semilinear equations, Comm. Partial Differential Equations 22 (1997), 661–684.
P.L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441–467.
P.L. Lions, Bifurcation and optimal stochastic control, Nonlinear Anal. 2 (1983), 177–207.
P.L. Lions, Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations I. The dynamic programming principle and applications, Comm. Partial Differential Equations 8 (1983), 1101–1174.
P.L. Lions, Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations II. Viscosity solutions and uniqueness, Comm. Partial Differential Equations 8 (1983), 1229–1276.
P.L. Lions, Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations III. Regularity of the optimal cost function, Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, vol. V (Paris, 1981/1982), 1983, pp. 95–205.
T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem, J. Differential Equations 146 (1998), 121–156.
T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem II, J. Differential Equations 158 (1999), 94–151.
C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pure Appl. 72 (1966), 141–170.
A. Quaas, Existence of positive solutions to a “semilinear” equation involving the Pucci’s operator in a convex domain, Differential Integral Equations 17 (2004), 481–494.
A. Quaas and A. Allendes, Multiplicity results for extremal operators through bifurcation, Discrete Contin. Dyn. Syst. 29 (2011), 51–65.
A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math. 218 (2008), 105–135.
A. Quaas and B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci operator, Comm. Partial Differential Equations 31 (2006), 987–1003.
P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.
P.H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal. 14 (1973), 462–475.
G.T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958.
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